There are two main types of mutation that I know: cluster (or quiver) mutations, and mutations of polytopes (roughly equivalent to mutations of Laurent polynomials). If you haven’t had the joy of meeting these yet, see here and here for introductions (only one of which was written by me). A natural aim is to reconcile these two related notions. To me at least, a reasonable approach is to try to find a cluster algebra whose seeds biject with polytopes in a given mutation class, perhaps by associating some integer invariants to them which can then appear in a quiver. You may recall from a few posts ago that this was essentially the strategy for communicating the Markov spectrum into the cluster vocabulary. Indeed, the Markov cluster algebra is going to be a constant companion here as we encounter it in a more geometrical guise: the Markov triples parameterise the del Pezzo surfaces that degenerate to via sending a triple to the weighted projective plane . One of the hopes of mirror symmetry is to find a cluster algebra (or cluster variety) that somehow gathers together all the toric degenerations of a given Fano variety, which is partly accomplished for by the Markov cluster algebra. As a result of this classification for – due to Hacking-Prokhorov – that was extended to describe mutations between arbitrary weighted projective planes by Akhtar-Kasprzyk, some deformation invariants naturally appear. The blurb is that to each mutation class of weighted projective planes, one can associate a Diophantine equation (generalising the Markov equation) whose integer solutions are the weights for weighted projective planes in that class. The coefficients of this equation are expressible in terms of the weight data or, more concretely, in combinatorial terms. It is also interesting to see how other related quantities change (or don’t change) as one traverses the mutation class. Throughout, I am using the principle that -Gorenstein deformations of toric Fanos correspond to mutations of the associated Fano polytopes. Anyway, my survey article can be found here and includes some comments about these deformation invariants that I don’t think you can find elsewhere.
As a last remark, let me describe how the smoothable and nonsmoothable cases differ with regard to their behaviour under mutations. One can mutate away -singularities (i.e. the smoothable ones) and a toric Fano is smoothable iff all of its singularities are s. In the case of weighted projective planes, which correspond to Fano triangles as discussed last time, if the wpp is smoothable then there are three possible mutations: one in each face. From Hacking-Prokhorov’s classification, it turns out that each of these -singularities is elementary; here, it means that the -singularity is as thin as possible and so only admits one mutation. The same is true of the resulting wpp after mutation. Hence, the graph of mutations is a 3-regular tree. In particular, all of the mutations are involutions. Suppose that a wpp has a nonelementary -singularity (which means as a result that it will be nonsmoothable). This means that multiple mutations in the same direction are possible, which is very dissimilar to cluster mutations. And, furthermore, the result of mutation might not even be a simplex anymore! These difficulties – and others like them, for instance when the singularities are residual so have no smoothable part – render the translation between the geometric and cluster worlds very nontrivial, making the special cases like where many things work… special.